| L(s) = 1 | + 2·3-s + 4·7-s − 4·9-s + 4·17-s + 4·19-s + 8·21-s − 3·23-s − 13·27-s − 4·29-s + 10·31-s − 10·37-s − 4·41-s − 24·43-s + 16·47-s − 49-s + 8·51-s + 10·53-s + 8·57-s − 11·59-s − 4·61-s − 16·63-s + 12·67-s − 6·69-s + 4·71-s + 4·73-s − 4·79-s + 4·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s − 4/3·9-s + 0.970·17-s + 0.917·19-s + 1.74·21-s − 0.625·23-s − 2.50·27-s − 0.742·29-s + 1.79·31-s − 1.64·37-s − 0.624·41-s − 3.65·43-s + 2.33·47-s − 1/7·49-s + 1.12·51-s + 1.37·53-s + 1.05·57-s − 1.43·59-s − 0.512·61-s − 2.01·63-s + 1.46·67-s − 0.722·69-s + 0.474·71-s + 0.468·73-s − 0.450·79-s + 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.740156258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.740156258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - 2 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_i_al |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 48 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_r_abw |
| 11 | $A_4\times C_2$ | \( 1 + 5 T^{2} + 56 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.11.a_f_ce |
| 13 | $A_4\times C_2$ | \( 1 + 32 T^{2} - 7 T^{3} + 32 p T^{4} + p^{3} T^{6} \) | 3.13.a_bg_ah |
| 17 | $A_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 128 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aey |
| 19 | $A_4\times C_2$ | \( 1 - 4 T + 25 T^{2} - 88 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ae_z_adk |
| 29 | $A_4\times C_2$ | \( 1 + 4 T + 48 T^{2} + 63 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.e_bw_cl |
| 31 | $A_4\times C_2$ | \( 1 - 10 T + 110 T^{2} - 579 T^{3} + 110 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ak_eg_awh |
| 37 | $A_4\times C_2$ | \( 1 + 10 T + 79 T^{2} + 412 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.k_db_pw |
| 41 | $A_4\times C_2$ | \( 1 + 4 T + 84 T^{2} + 9 p T^{3} + 84 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_dg_of |
| 43 | $A_4\times C_2$ | \( 1 + 24 T + 293 T^{2} + 2296 T^{3} + 293 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.y_lh_dki |
| 47 | $A_4\times C_2$ | \( 1 - 16 T + 210 T^{2} - 1587 T^{3} + 210 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.aq_ic_acjb |
| 53 | $A_4\times C_2$ | \( 1 - 10 T + 155 T^{2} - 956 T^{3} + 155 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ak_fz_abku |
| 59 | $A_4\times C_2$ | \( 1 + 11 T + 152 T^{2} + 919 T^{3} + 152 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.l_fw_bjj |
| 61 | $A_4\times C_2$ | \( 1 + 4 T + 67 T^{2} + 592 T^{3} + 67 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.e_cp_wu |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 137 T^{2} - 776 T^{3} + 137 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_fh_abdw |
| 71 | $A_4\times C_2$ | \( 1 - 4 T + 62 T^{2} - 35 T^{3} + 62 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ae_ck_abj |
| 73 | $A_4\times C_2$ | \( 1 - 4 T + 68 T^{2} - 933 T^{3} + 68 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ae_cq_abjx |
| 79 | $A_4\times C_2$ | \( 1 + 4 T + 121 T^{2} + 64 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.e_er_cm |
| 83 | $A_4\times C_2$ | \( 1 + 8 T + 9 T^{2} - 528 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.i_j_aui |
| 89 | $A_4\times C_2$ | \( 1 + 20 T + 335 T^{2} + 3568 T^{3} + 335 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_mx_fhg |
| 97 | $A_4\times C_2$ | \( 1 - 38 T + 763 T^{2} - 9284 T^{3} + 763 p T^{4} - 38 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.abm_bdj_antc |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54922650627685580246993197356, −7.42498444963041742657241760771, −6.82410777421114822804931027757, −6.75282584271519609260758219095, −6.25373295417899084681489170106, −6.09977662577905792800304337577, −6.03294413307097334628500508136, −5.35996569166983094883843315367, −5.34025208001290531238620795452, −5.16508678551875178399171338200, −5.05831940585531226117667534419, −4.53878939064355490826793122550, −4.44925831557972388023322319980, −3.80676244996758125839049170237, −3.61740725387602433086897239574, −3.59637800006343769716525804954, −3.06767991439790270257907761089, −2.89399246098083307640500674180, −2.79097164746481690916482024466, −2.08547375876196695026451774694, −2.08184726446362094521438553331, −1.65851092607572085643866898772, −1.41865397833624484071521130682, −0.78859558304663548036663875697, −0.37328064475887228071451918765,
0.37328064475887228071451918765, 0.78859558304663548036663875697, 1.41865397833624484071521130682, 1.65851092607572085643866898772, 2.08184726446362094521438553331, 2.08547375876196695026451774694, 2.79097164746481690916482024466, 2.89399246098083307640500674180, 3.06767991439790270257907761089, 3.59637800006343769716525804954, 3.61740725387602433086897239574, 3.80676244996758125839049170237, 4.44925831557972388023322319980, 4.53878939064355490826793122550, 5.05831940585531226117667534419, 5.16508678551875178399171338200, 5.34025208001290531238620795452, 5.35996569166983094883843315367, 6.03294413307097334628500508136, 6.09977662577905792800304337577, 6.25373295417899084681489170106, 6.75282584271519609260758219095, 6.82410777421114822804931027757, 7.42498444963041742657241760771, 7.54922650627685580246993197356
